Mathematical Morphology Random sets and Porous Media dominique.jeulin@mines-paristech.fr, s et Systèmes Centre des Matériaux P.M. Fourt, Mines-ParisTech 1 L = 1 mm Origin Motivations G. Matheron, 1967 Characterization of the morphology of a heterogeneous medium? Prediction of the macroscopic behaviour of a porous medium (composition of permeabilities)? Representation of a heterogeneous medium by a model? 2 Introduction 3 4 Extraction of quantitative information on microstructures (3D images, measurements) Models of random sets and simulation of microstructures Theory of random sets and tools to solve homogenization problems Characterization of a random structure 3D Images Morphological Criteria Probabilistic Criteria 3D Images of Intermetallic particles in Al alloys X ray microtomography performed at the ESRF (ID 19 line) 3D measurements High resolution (0.7μm) 3D complex shapes of particles (morphological parameters, local curvature) (E. Parra-Denis PhD) 5 ZnO by Nanotomography ZnO needles after sublimation in the solair oven PROMES Laboratory (CNRS Odeillo) Application : ionic conduction Acquisition : -42 à +37 with 1 increments Resolution of images : x12000 2 nm 1 voxel 6 Arnaud GROSJEAN Dominique JEULIN Maxime MOREAUD Alain THOREL 1
ZnO by Nanotomography 7 8 Resolution : 2 nm 1 voxel 9 Characterization of a random structure: Main Criteria Characterization of a random set 10 Morphological criteria Size Shape Distribution in space (Clustering, Scales, Anisotropy) Connectivity Probabilistic criteria Probability laws (n points, sup K ) Moments Models derived from the theory of Random Sets by G. MATHERON For a random closed set A (RACS), characterization by the CHOQUET capacity T(K) defined on the compact sets K c T( K) = P{ K I A Φ } = 1 P{ K A } = 1 Q( K) In the euclidean space R n, CHOQUET capacity and dilation operation T( K x) = P { x A K } Binary Morphology (Fe-Ag) 11 Basic Operations of MM 12 L = 250 µm Erosion hexagonal (2) Dilation hexagonal (2) 2
Morphological interpretation 13 Calculation of the CHOQUET capacity 14 Experimental estimation of T(K) by image analysis, using realizations of A, and dilation operation. General case: several realizations and estimation for every point x For a stationary random set, T(K x ) = T(K); For an ergodic random set, T(K) estimated from a single realization by measurement of a volume fraction Every compact set K (points, ball...) brings its own information on the random set A For a given model, the functional T is obtained: by theoretical calculation by estimation on simulations on real structures (possible estimation of the parameters from the "experimental" T, and tests of the validity of assumptions). 15 Point Processes 16 Models of Random Structures Most simple kind of random structure: very small defects isolated in a matrix Particular RACS: Choquet capacity T(K) Probability generating function G K (s) of the random variable N(K) (number of points of the process contained in K) Poisson Point Process Prototype random process without any order 17 18 Random sets and Random Functions Models Starting from a point process, more general models, called grain models: The Boolean model The dead leaves model Random function models 3
Boolean Model 19 Boolean Model (G. Matheron, 1967) 20 The Boolean model (G. Matheron) is obtained by implantation of random primary grains A (with possible overlaps) on Poisson points x k with the intensity q: A = U xk A xk Any shape (convex or non convex, and even non connected) can be used for the grain A ' Fe-Ag Alloy Boolean Model of Spheres (0.5) Boolean Model of Spheres 3D Simulation 21 Hard Spheres 3D Simulation 22 D. Jeulin D. Jeulin M. Faessel M. Faessel (CMM) (CMM) 23 2 scales Cox Boolean model of Spheres 3D Simulation Boolean Model 24 D. Jeulin M. Faessel (CMM) WC Co (J.L. Chermant, M. Coster, J.L. Quennec h et D. Jeulin) L = 40 µm Poisson Boolean Model (P. Delfiner) 4
Choquet capacity, with T Boolean Model q = P Ex: contact distribution (ball), covariance c { x A } μn( A' ( θ μn( A' K) ) = 1 q μn( ') K) ( K) = 1 Q( K) = 1 exp A Percolation threshold obtained from simulations: 0:2895 +- 0:0005 for spheres with a single diameter Percolation threshold related to the zeroes of N V G V (p) 25 Percolation threshold Materials made of components with a high contrast of properties: strong effect on the macroscopic properties when a given phase (e.g. the pores) percolates through the structure (connected paths in the samples of the medium) For a given model, estimation of the critical percolation threshold ρc (volume fraction above which a component percolates) 26 Percolation of random structures 27 How to estimate a percolation threshold? 28 Labelisation of aggregates allows easy extraction of aggregates connecting two opposite faces No percolation : no path connects two opposite faces Percolation: one aggregate connects two opposite faces Percolation threshold ρc = volume fraction of objects when 50% of the realizations have aggregates connecting two opposite faces Carbon nanotubes composite materials Outstanding mechanical, electrical or chemical properties, mainly due to their low percolation threshold 29 Percolation of a Boolean model of spheres [*] Jeulin D., Moreaud M. Multi-scale simulation of random spheres aggregates application to nanocomposites, Proc. 9th European Congress on Stereology and Image Analysis, Zakopane, Poland, May 10-13 2005, Vol. I p. 341-348. Volume : 2000 3 Average Nb.. of simulated spheres about 5 000 000 sphere radius : 10 Average computation time for one realization 48 s. 30 Percolation threshold 0.2897±0.0004 0.0004 PC PIV 2.6GHz RAM 768 Mo (*) 0.2895±0.005 0.005 M.D. Rintoul et S. Torquato, Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model, J. Phys. A: Math. Gen. 30, L585-L592, L592, 1997 5
Percolation of a Boolean model of spheres (complementary set) 31 32 Percolation of a Boolean model of spherocylinders with uniform orientations Estimation of the percolation threshold of the complementary random set of a boolean model of spheres (constant radius) One scale simulation Periodic simulations along axis x and y (50 realizations) AR=l / r V=300 3, l=50, r=2, Vv=0.02 ρc = 0.0540±0.005 0.005 Aspect ratio l/r 3000 500 100 ρ c 0.00037 0.00232 0.01145 0.09780 Jeulin D., Moreaud M. Percolation of multi-scale fiber aggregates, Proc. 6th International Conference on Stereology, Spatial Statistics and Stochastic Geometry, Czech Republic, June 26-26, 2006. 10 Percolation of oriented spherocylinders 33 Percolation of oriented spherocylinders 34 Materials like papers or carbon nanotubes enhanced elastomers are composed of oriented fibers φ between x and +x V=300 3, l=50, r=2, Vv=0.02, x=± 15 Simulation of oriented sphero-cylinders: Φ uniformly distributed, φ limited between x and +x More oriented the fibers are, higher ρc is in the two directions (axis Z and plane XY) Combination of basic random sets 35 Intersection of random sets 36 Starting from the basic models, more complex structures, such as superposition of scales, or fluctuations of the local volume fraction p of one phase Union or intersection of independent random sets A = I A1 A2 { ( A1 A2 )} = P{ K A1 } P{ K 2} P( K) = P K I A P = 0.49 P = 0.49 (P 1 = P 2 =0.7) 6
37 Nanocomposite carbon black - polymer (TEM) 38 Simulation of a Carbon black Nanocomposite Transmission micrographs (L. Savary, D. Jeulin, A. Thorel) Intersection of 3 scales Boolean models of spheres (identification from thick sections) Percolation of multiscale aggregates Cox Boolean model 39 Percolation multiscale spherical aggregates Two scales of spheres 40 Volume : 2000 3 sphères : 10 aggregates : 300 Simulation Labeling of aggregates s 2 = 0.0838 (s: ρc c SB spheres) ρc = 0.085±0.003 0.003 Critical percolation threshold lower than for the Boolean model of spheres (0.2895) [*] Jeulin D., Moreaud M. Multi-scale simulation of random spheres aggregates application to nanocomposites, Proc. 9th European Congress on Stereology and Image Analysis, Zakopane, Poland, May 10-13 2005, Vol. I p. 341-348. Percolation of multi-scale distributions of sphero-cylinders Two-scale simulation, with sphero-cylinders randomly located into inclusion spheres according to a Cox point process Dichotomic research used to estimate ρc with 20 realizations for each volume fraction ρc = 0.05% V=300 3, l=30, r=2, Vv=0.05 41 V=300 3, r=60, Vv=0.3 Percolation of multi-scale distributions of sphero-cylinders Aspect ratio AR 1000 Volume dimensions = (5 x D) 3 Diameter of spheres D 2 x AR Vv of spheres 0.51 0.44 0.34 ρ c 0.00051 0.00047 0.00049 Aspect ratio AR 50 Volume dimensions = (5 x D) 3 Diameter of spheres D 2 x AR 5 x AR Vv of spheres 0.51 0.42 0.35 0.5 0.43 0.35 ρ c 0.014 0.016 0.019 0.014 0.012 0.012 ρc lower than for a homogenous distribution lowest ρc obtained for a large diameter of spheres and a low volume fraction of spheres 42 7
Color dead leaves (D. Jeulin, 1979) 43 Color dead leaves: non overlaping grains (D. Jeulin, 1997) 44 Size distribution (initial intact); single symmetric convex grain in R n for homogeneous model: V V < = 1 / 2 n Random Functions Models 45 Boolean random functions 46 Continuous version of random sets models Boolean RF: random implantation of primary random functions on points of a Poisson point process. TheU operation for overlapping grains is replaced by the supremum or by the infimum Change of support by Sup or by Inf (Extreme Values) Cone Primary grains Boolean Variety Powders and dead leaves 47 Reaction-Diffusion Models 48 Reaction-Diffusion Thesis L. Decker (1999) D. Jeulin, CALGON L = 15.5 µm Turing texture Simulation of connected media 8
Change of scale in random media (Physics-Texture) 49 Homogenization 50 From Nano to Macro Heterogeneous Medium (composite, porous medium, metalli polycristal, rocks, biological medium, rough surfaces ) Eqyuvalent Homogeneous Medium? Prediction of the «effective» properties Examples of homogenization problems Thermal conductivity of a two-component medium (thermal insulation) Transport properties of porous media Elastic Moduli of heterogeneous medium Wave Propagation in heterogeneous medium (electromagnetic, acoustic ) Optical Properties of a heterogeneous medium or of a rough surface 51 Examples of Physical Properties Problem Thermal conduction Electrostatics Elasticity Fluid Flow in porous media Heat Flux Dielectric Displacement Stress Velocity Temperature Gradient Electric Field Srain Pressure Gradient Property Thermal Conductivity Dielectric Permittivity Elastic Moduli Permeability 52 Homogenization Bounds of macroscopic properties (order 3: multicomponent random sets and functions) Optimal random Microstructures Probabilistic Definition of the RVE for numerical simulations 53 Change of scale Applications of the models of random media to the prediction of the macroscopic behavior of a physical system from its microscopic behavior. Estimation of the effective properties (overall properties of an equivalent homogeneous medium) of random heterogeneous media from their microstructure (Homogenization) From variational principles, bounds of the effective properties for linear constitutive equations. Estimation of the effective behaviour from numerical simulations on random media. Fracture statistics models 54 9
Thermal Conductivity of Ceramics (Use of Bounds for the Boolean Model) Textures AlN(λ=100) with a Y rich binder (λ= 10) C. Pélissonnier, D. Jeulin, A. Thorel 55 Thermal Conductivity of Ceramics: 3rd Order Bounds of the Boolean model 56 57 Random Composite with optimal effective properties 58 Digital Materials Digital Materials 59 Homogenization and Simulation 60 Input of 3D images Real images (confocal, microtomography) Simulations from a random model Use of a computational code (Finite Elements, Fast Fourier Transform, PDE numerical solver) Homogenization Representative Volume Element? Prediction of the effective properties by 3D FFT Nano Composites Carbon Black - polymer: permittivity, from a multi-scale random model (PhD A. Delarue, 2001 M. Moreaud, 2007, D. Jeulin, A. Thorel, DGA, EADS) Charged Elastomers: elastic behaviour (PhD A. Jean, Michelin, 2006-2009 (19 February)) 10
Method: Homogenization and Simulation Properties of components + Iterative algorithm based on Fourier tansform, to solve the Gauss equation of electrostatics, derived from the Maxwell equations Periodic boundary conditions Field D(x) in each point Field E(x) in each point ε*= D E 61 L Ice cream: mechanical behavior (Unilever) Three scales of observation d << l << (thesis T. Kanit, 2003, S. Forest) RVE d l L 62 Morphology and effectives properties *Heterogeneities in 2D 63 Morphology and Young s modulus Experimental measurements (4-point bending test) 64 100 µm Coarse microstructure Fine microstructure How to predict the Young s modulus for different microstructures? Analytical models: bounds and 65 bounds and estimations Principle of Homogenization Theory 66 * Bounds : very large. *Estimation : does not really take the morphology into account. Not really useful for media with a high contrast in properties Two-phase heterogeneous material with elastic tensors C 1 and C 2 Homogeneous equivalent material with the macroscopic elastic tensor C eff We have: With the spatial averages Σ E =C eff Where σ and ε Σ=< σ > E =< ε > 1 < P >= P ( x ) dx V V are the local stress and strain tensors 11
Principle of Homogenization Theory 67 RVE and Integral Range Two-phase heterogeneous material with elastic tensors C 1 and C 2 Average elastic energy (Hill-Mandel lemma) ) of a specimen V submitted to one of the following boundary conditions: - KUBC : kinematic (strain) uniform boundary conditions - SUBC : static (stress) uniform boundary conditions - PERIODIC : periodic boundary conditions σ and < σ: ε>=< ε: C: ε>=σ: E=< ε> < ε> and ε are uncorrelated C eff P(x): : local random property (indicator function, Young s modulus,, ) in the domain V. Local variance of P: D 2 [P(x) P(x)] Mean value of P in V: < P >= P x dx V 1 ( ) V 3 D Variance of the mean: 2 [ < P> ] = D2[ P( x )] V A A 3 is the Integral Range of P (giving the size of a RVE for P) If A 3 << V, V may be subdivided into N = V /A 3 subdomains with uncorrelated properties P i Integral Range and Covariance 69 Integral Range and Covariance 70 Fluctuations of the average values <P> of P(x) (stationary RF) in the domain V, as a function of the centred covariance W 2 h W2( h) = E{( P( x+ h) E( P))( P( x) E( P)} 2 1 D ( V) = W ( x y dxdy P 2 V ) 2 V V For large specimens (V >> A 3 ), where A 3 is the integral range, asymptotic formula for the variance: 2 2 A3 DP ( V ) = DP V 1 with A3 = W 2( h) dh 2 3 D R P where D 2 P is the point variance of P(x) Set Covariance 71 Microstructure and Covariance 72 Covariance C(h) of a random set A without AFP C x,x h P x A,x h A For a stationary random set, C(x,x+h) = C(h) For an ergodic random set, C(h) is estimated by the volume fraction of A A h with AFP 12
Set Covariance and field Covariance 73 74 Determination of the integral range and of the RVE Connexion between the two? Generally, no direct link!! The Covariance of fields depends on all «sets» moments with n points Start from realisations of the microstructure (images or simulations) Use appropriate Boundary Conditions (Periodic, ) to estimate the effectives properties of every realisation Estimate the average and the variance of effectives properties as a function of the volume of specimens Estimate A 3, the RVE and the number of fields to simulate as a function of the wanted precision Elastic Moduli and Thermal Conductivity of the 3D Voronoï Mosaïc 75 Thermal computation on Voronoï mosaïc 70% ice, contrast = 100 in thermal conductivity Computation of thermal conductivity in 3 directions 76 3D Voronoï space tessellation: zones of inluence of random Poisson points Independent random coloration of each Voronoï polyhedron: Poisson mosaïc (approximation of some real two-phase textures) Finite element calculations on finite volume realizations of Voronoï mosaïc in V map of the flux in direction (z) with periodic b. c. (8281 d.o.f.) map of the temperature with UGT b. c. (312481 d.o.f. on 20 pc) Results for the Voronoï mosaïc * Fluctuations of the thermal conductivity -volume fraction = 70% of hard phase (ice( ice) -contrast in thermal conductivity = 100 -UGT : uniform temperature gradient at the boundary -UHF : uniform heat flux at the boundary -PERIODIC : periodic boundary conditions 77 Results for the Voronoi mosaïc Minimal number of realizations for periodic boundary conditions Example: : RVE for unbiased volume = 125 grains ε relative = 1% ε relative = 5% 78 thermal conductivity number of RVE 765 number of RVE 30 13
Integral range (volume fraction P) 79 Integral range (elastic moduli) 80 0.5 1.178 E 1 / E 2 = 100 P = 0.7 1.111 K (P = 0.7) KUBC 2.088 0.9 1.177 µ (P = 0.7) KUBC 1.863 Gilbert (1962): 1.179 µ (P = 0.5) Périodic 1.637 Integral range (thermal conductivity) 81 82 P = 0.7; λ 1 / λ 2 = 100 UGT 2.335 Integral Range and Representative Volume Element for the estimation of the Elastic Moduli of the Boolean Model of Spheres λ UHF 2.036 (joint work with F. Willot) Périodic 2.619 Elastic Fields solved by the Fourier transform (FFT) Algorithm based on the Lippmann-Shwinger equations (Moulinec, Suquet, 1994). Introduction of a homogeneous elastic tensor L (0) (reference medium) and of the corresponding Green function G (0) Iterations in the Fourier space (Green function) and in the real space. For G (0) with zeo mean: 83 Effective Bulk Modulus of the Boolean model of spheres 84 ĸ µ matrice 1/3 ½ inclusions 1000 1000 Periodic boundary conditions. Operation on images, without any mesh Convergence for an infinite contrast with the improved algorithm increased Lagrangien (Moulinec, Suquet + Moulinec, 2001) For an isotropic medium, in the Fourier space: p c Bulk Modulus as a function of the concentration of the rigid phase and HS, Beran bounds f=0,2 14
Effective Bulk Modulus of the Boolean model of spheres 85 pores Non compressible matrix (Poisson coeff. 0,4999) Integral Range for different properties 86 Sphere V V Sphere V V Bulk Modulus Int. J. of Eng. Sc. 2008, in press Integral range A 3 Representative Volume Element 87 Conclusion 88 The RVE depends on: The morphology The physical property (elastic moduli, thermal conductivity, dielectric permittivity, ) The contrast between properties of components The type of boundary conditions (uniform, periodic,;..) Random models of structures, to simulate the complex morphology of microstructures Approach, based on measurements obtained by image analysis: test and select appropriate models, estimate their parameters Physical and Morphological Modelling «Which textures for which properties?» Possible use in the synthesis of textures Many domains of application: Materials, Nano composites, Porous media, Biology, Food, Surfaces, Vision... To learn more Courses ENSMP, 60 Bd Saint-Michel, Paris Physics and Mechanics of Random Media (30 March - 3 April 2009) Models of Random Structures (November( 2009) Groupe de Travail MECAMAT «Approches probabilistes en Mécanique des Milieux Hétérogènes» (Bordeaux, 14-15 May 2009) http://cmm.ensmp.fr/ 89 15